Helical surfaces with a constant ratio of principal curvatures

Yang Liu; Olimjoni Pirahmad; Hui Wang; Dominik L. Michels; Helmut Pottmann

doi:10.1007/s13366-022-00670-y

Helical surfaces with a constant ratio of principal curvatures

Yang Liu, Olimjoni Pirahmad, Hui Wang, Dominik L. Michels, Helmut Pottmann^*

^*Corresponding author for this work

Research output: Contribution to journal › Article › peer-review

2 Scopus citations

Abstract

We determine all helical surfaces in three-dimensional Euclidean space which possess a constant ratio a: = κ₁/ κ₂ of principal curvatures (CRPC surfaces), thus providing the first explicit CRPC surfaces beyond the known rotational ones. Our approach is based on the involution of conjugate surface tangents and on well chosen generating profiles such that the characterizing differential equation is sufficiently simple to be solved explicitly. We analyze the resulting surfaces, their behavior at singularities that occur for a> 0 , and provide an overview of the possible shapes.

Original language	English (US)
Pages (from-to)	1087-1105
Number of pages	19
Journal	Beitrage zur Algebra und Geometrie
Volume	64
Issue number	4
DOIs	https://doi.org/10.1007/s13366-022-00670-y
State	Published - Dec 2023

Bibliographical note

Publisher Copyright:
© 2022, The Managing Editors.

Keywords

Helical surface
Surface with a constant ratio of principal curvatures
Weingarten surface

ASJC Scopus subject areas

Algebra and Number Theory
Geometry and Topology

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10.1007/s13366-022-00670-y

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title = "Helical surfaces with a constant ratio of principal curvatures",

abstract = "We determine all helical surfaces in three-dimensional Euclidean space which possess a constant ratio a: = κ1/ κ2 of principal curvatures (CRPC surfaces), thus providing the first explicit CRPC surfaces beyond the known rotational ones. Our approach is based on the involution of conjugate surface tangents and on well chosen generating profiles such that the characterizing differential equation is sufficiently simple to be solved explicitly. We analyze the resulting surfaces, their behavior at singularities that occur for a> 0 , and provide an overview of the possible shapes.",

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author = "Yang Liu and Olimjoni Pirahmad and Hui Wang and Michels, {Dominik L.} and Helmut Pottmann",

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doi = "10.1007/s13366-022-00670-y",

language = "English (US)",

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T1 - Helical surfaces with a constant ratio of principal curvatures

AU - Liu, Yang

AU - Pirahmad, Olimjoni

AU - Wang, Hui

AU - Michels, Dominik L.

AU - Pottmann, Helmut

PY - 2023/12

Y1 - 2023/12

N2 - We determine all helical surfaces in three-dimensional Euclidean space which possess a constant ratio a: = κ1/ κ2 of principal curvatures (CRPC surfaces), thus providing the first explicit CRPC surfaces beyond the known rotational ones. Our approach is based on the involution of conjugate surface tangents and on well chosen generating profiles such that the characterizing differential equation is sufficiently simple to be solved explicitly. We analyze the resulting surfaces, their behavior at singularities that occur for a> 0 , and provide an overview of the possible shapes.

AB - We determine all helical surfaces in three-dimensional Euclidean space which possess a constant ratio a: = κ1/ κ2 of principal curvatures (CRPC surfaces), thus providing the first explicit CRPC surfaces beyond the known rotational ones. Our approach is based on the involution of conjugate surface tangents and on well chosen generating profiles such that the characterizing differential equation is sufficiently simple to be solved explicitly. We analyze the resulting surfaces, their behavior at singularities that occur for a> 0 , and provide an overview of the possible shapes.

KW - Helical surface

KW - Surface with a constant ratio of principal curvatures

KW - Weingarten surface

UR - http://www.scopus.com/inward/record.url?scp=85144667163&partnerID=8YFLogxK

U2 - 10.1007/s13366-022-00670-y

DO - 10.1007/s13366-022-00670-y

M3 - Article

AN - SCOPUS:85144667163

SN - 0138-4821

VL - 64

SP - 1087

EP - 1105

JO - Beitrage zur Algebra und Geometrie

JF - Beitrage zur Algebra und Geometrie

IS - 4

ER -

Helical surfaces with a constant ratio of principal curvatures

Abstract

Bibliographical note

Keywords

ASJC Scopus subject areas

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